Response Spectrum Analysis

Response spectrum analysis can be used to estimate the peak response (displacement, stress, etc.) of a structure to a particular base motion or force. The method is only approximate, but it is often a useful, inexpensive method for preliminary design studies.

The response spectrum procedure is based on using a subset of the modes of the system, which must first be extracted by using the eigenfrequency extraction procedure. The modes will include eigenmodes and, if activated in the eigenfrequency extraction step, residual modes. The number of modes extracted must be sufficient to model the dynamic response of the system adequately, which is a matter of judgment on your part.

If the number of eigenmodes included in the superposition does not sufficiently represent the total mass of the structure, you can use the missing mass method to augment the missing inertia in the dynamic response as described in Using the Missing Mass Method.

In cases with repeated eigenvalues and eigenvectors, the modal summation results must be interpreted with care. You should use mode combination rules that are appropriate for closely spaced modes.

While the response in the response spectrum procedure is for linear vibrations, the prior response may be nonlinear. Initial stress effects (stress stiffening) will be included in the response spectrum analysis if nonlinear geometric effects (General and Perturbation Procedures) were included in a general analysis step prior to the eigenfrequency extraction step.

The problem to be solved can be stated as follows: given a set of base motions, ${\ddot{u}}_{j}^{B} (t)$ ( $j = 1, 2, 3$ ), specified in orthogonal directions defined by direction cosines $t_{k}^{j}$ ( $k = 1, 2, 3$ ), estimate the peak value over all time of the response of any variable in a finite element model that is simultaneously subjected to these multiple base motions. The peak response is first computed independently for each direction of excitation for each natural mode of the system as a function of frequency and damping. These independent responses are then combined to create an estimate of the actual peak response of any variable chosen for output, as a function of frequency and damping.

The acceleration history (base motion) is not given directly in a response spectrum analysis; it must first be converted into a spectrum.

Specifying a Spectrum

The response spectrum method is based on first finding the peak response to each base motion excitation of a one degree of freedom system that has a natural frequency equal to the frequency of interest. The single degree of freedom system is characterized by its undamped natural frequency, $ω_{α}$ , and the fraction of critical damping present in the system, $ξ_{α}$ , at each mode $α$ . The equations of motion of the system are integrated through time to find peak values of relative displacement, relative velocity, and relative or absolute acceleration for the linear, one degree of freedom system. This process is repeated for all frequency and damping values in the range of interest. Plots of these responses are known as displacement, velocity, and acceleration spectra: $S^{D} (ω_{α}, ξ_{α})$ , $S^{V} (ω_{α}, ξ_{α})$ , and $S^{A} (ω_{α}, ξ_{α})$ . The response spectrum can be obtained directly from measured data, as described in Defining a Spectrum Using Values of S as a Function of Frequency and Damping below. You can also use a Fortran program to define a spectrum; an example of defining a spectrum from an acceleration record in this way is provided in Analysis of a cantilever subject to earthquake motion.

Alternatively, you can create the required spectrum by specifying an amplitude (time history record), the frequency range, and the damping values for which the spectrum will be built, as described in Creating a Spectrum from a Given Time History Record below. The spectrum can be used in the subsequent response spectrum analysis, or it can be written to a file for future use.

For each damping value the magnitude of the response spectrum must be given over the entire range of frequencies needed, in ascending value of frequency. Abaqus/Standard interpolates linearly between the values given on a log-log scale. Outside the extremes of the frequency range given, the magnitude is assumed to be constant, corresponding to the end value given. (See Material Data Definition for an explanation of data interpolation.)

Any number of spectra can be defined, and each spectrum must be named. The response spectrum procedure allows up to three spectra to be applied simultaneously to the model in orthogonal physical directions defined by their direction cosines.

Defining a Spectrum Using Values of S as a Function of Frequency and Damping

You can define a spectrum by specifying values for the magnitude of the spectrum; frequency, in cycles per time, at which the magnitude is used; and associated damping, given as a ratio of critical damping.

Input File Usage

To define the spectrum on the data lines:

SPECTRUM, NAME=spectrum name

Repeat this option to define multiple spectra for an analysis.

Abaqus/CAE Usage

To define a spectrum, do the following:

Step, Interaction, or Load module: ToolsAmplitudeCreate; Name: spectrum name, Type: Spectrum

To apply a spectrum to the model, do the following:

Step module: Create Step: Linear perturbation: Response spectrum: Use response spectrum: select spectrum name for each physical direction in which it should be applied

Specifying the Type of Spectrum

You can indicate whether a displacement, velocity, or acceleration spectrum is given. The default is an acceleration spectrum.

Alternatively, an acceleration spectrum can be given in g-units. In this case you must also specify the value of the acceleration of gravity.

Input File Usage

Use one of the following options to define a displacement, velocity, or acceleration spectrum:

SPECTRUM, NAME=name, TYPE=DISPLACEMENT
SPECTRUM, NAME=name, TYPE=VELOCITY
SPECTRUM, NAME=name, TYPE=ACCELERATION

Use the following option to define an acceleration spectrum given in g-units:

SPECTRUM, NAME=name, TYPE=G, G=g

Abaqus/CAE Usage

Use one of the following options to define a displacement, velocity, or acceleration spectrum:

Step, Interaction, or Load module: ToolsAmplitudeCreate; Type: Spectrum; Specification units: Displacement, Velocity, or Acceleration

Use the following option to define an acceleration spectrum given in g-units:

Step, Interaction, or Load module: ToolsAmplitudeCreate; Type: Spectrum; Specification units: Gravity, Gravity: g

Reading the Data Defining the Spectrum from an Alternate Input File

The data for the spectrum can be specified in an alternate input file and read into the Abaqus/Standard input file.

Input File Usage

SPECTRUM, NAME=name, INPUT=file name

Abaqus/CAE Usage

Step, Interaction, or Load module: ToolsAmplitudeCreate; Type: Spectrum; click mouse button 3 while holding the cursor over the data table, and select Read from File

Creating a Spectrum from a Given Time History Record

If you have a time history of a dynamic event (e.g., acceleration, velocity, displacement), you can build your own spectrum by specifying the record type and the amplitude name that this record represents. If the amplitude record is given with an arbitrarily changing time increment, linear interpolation will be needed for the implicit integration scheme for the dynamic equation of motion for a single degree of freedom system subjected to this record. You can specify the frequency range for the integration scheme and the frequency increment. You can build a spectrum for every fraction of critical damping indicated in the list of damping values.

Input File Usage

SPECTRUM, CREATE, AMPLITUDE=amplitude name, 
NAME=spectrum name, TIME INCREMENT=dt

Abaqus/CAE Usage

Creating a spectrum from a given time history record is not supported in Abaqus/CAE.

Specifying the Type of Spectrum to Be Created

You can indicate whether a displacement, velocity, or acceleration spectrum is to be created. The default is an acceleration spectrum.

Alternatively, an acceleration spectrum can be created in g-units. In this case you must also specify the value of the acceleration of gravity.

Input File Usage

Use one of the following options to create a displacement, velocity, or acceleration spectrum:

SPECTRUM, CREATE, TYPE=DISPLACEMENT
SPECTRUM, CREATE, TYPE=VELOCITY
SPECTRUM, CREATE, TYPE=ACCELERATION

Use the following option to create an acceleration spectrum in g-units:

SPECTRUM, CREATE, TYPE=G, G=g

Abaqus/CAE Usage

Creating a spectrum from a given time history record is not supported in Abaqus/CAE.

Specifying the Record Type That the Time History Represents

You can indicate whether a displacement, velocity, or acceleration amplitude is specified. The default is an acceleration amplitude.

Alternatively, an acceleration amplitude can be given in g-units. In this case you must also specify the value of the acceleration of gravity.

Input File Usage

Use one of the following options to indicate that the amplitude is defined in displacement, velocity, or acceleration units:

SPECTRUM, CREATE, EVENT TYPE=DISPLACEMENT
SPECTRUM, CREATE, EVENT TYPE=VELOCITY
SPECTRUM, CREATE, EVENT TYPE=ACCELERATION

Use the following option to indicate that an acceleration amplitude is given in g-units:

SPECTRUM, CREATE, EVENT TYPE=G, G=g

Abaqus/CAE Usage

Creating a spectrum from a given time history record is not supported in Abaqus/CAE.

Creating an Absolute or Relative Acceleration Spectrum

When you create an acceleration spectrum from a given time history record, you can create an absolute or relative response spectrum. The default is an absolute spectrum.

Input File Usage

SPECTRUM, CREATE, TYPE=ACCELERATION,  ABSOLUTE
SPECTRUM, CREATE, TYPE=ACCELERATION,  RELATIVE

Abaqus/CAE Usage

Creating a spectrum from a given time history record is not supported in Abaqus/CAE.

Generating the List of Damping Values for the Fraction of Critical Damping

You must provide a list of damping values for the fraction of critical damping to create a spectrum. However, if the damping is evenly spaced between its lower and upper bound, you can automatically generate the list of damping values by providing the start value, end value, and increment for the fraction of critical damping.

Input File Usage

SPECTRUM, CREATE, DAMPING GENERATE

Abaqus/CAE Usage

Creating a spectrum from a given time history record is not supported in Abaqus/CAE.

Writing the Generated Spectra to an Independent File

You can write the generated spectra to an independent file. Otherwise, the generated spectra can be used only within the currently submitted job in subsequent response spectra procedures. You can inspect the generated spectra if you request that model definition data be printed to the data file (see Model and History Definition Summaries).

Input File Usage

SPECTRUM, CREATE, FILE=file name

Abaqus/CAE Usage

Creating a spectrum from a given time history record is not supported in Abaqus/CAE.

Estimating the Peak Values of the Modal Responses

Since the response spectrum procedure uses modal methods to define a model's response, the value of any physical variable is defined from the amplitudes of the modal responses (the “generalized coordinates”), $q_{α}$ . The first stage in the response spectrum procedure is to estimate the peak values of these modal responses. For mode $α$ and spectrum k this is

{(q_{α}^{m a x})}_{k} = c_{k} S_{k}^{D} (ω_{α}, ξ_{α}) \sum_{j} t_{j}^{k} Γ_{α j},

where

$q_{α}$: is the modal amplitude for mode $α$ ;
$c_{k}$: is a scaling parameter introduced as part of the response spectrum procedure definition for spectrum $S_{k}^{D} (ω_{α}, ξ_{α})$ ;
$S_{k}^{D} (ω_{α}, ξ_{α})$: is the user-defined value of the spectrum (see Specifying a Spectrum) in direction k interpolated, if necessary, at natural frequency $ω_{α}$ and the fraction of critical damping $ξ_{α}$ in mode $α$ ;
$t_{j}^{k}$: is the jth direction cosine for the kth spectrum; and
$Γ_{α j}$: is the participation factor for mode $α$ in direction j (see Natural Frequency Extraction).

Similar expressions for ${({\dot{q}}_{α}^{m a x})}_{k}$ and ${({\ddot{q}}_{α}^{m a x})}_{k}$ can be obtained by substituting velocity or acceleration spectra in the above equation.

Combining the Individual Peak Responses

The individual peak responses to the excitations in different directions will occur at different times and, therefore, must be combined into an overall peak response. Two combinations must be performed, and both introduce approximations into the results:

The multidirectional excitations must be combined into one overall response. This combination is controlled by the directional summation method, as described below in Directional Summation Methods.
The peak modal responses must be combined to estimate the peak physical response. This combination is controlled by the modal summation method, as described below in Modal Summation Methods.

Depending on the type of base excitation, either modal responses or directional responses are combined first.

Directional Summation Methods

You choose the method for combining the multidirectional excitations depending on the nature of the excitations.

The Algebraic Method

If the input spectra in the different directions are components of a base excitation that is approximately in a single direction in space, then for each mode the peak responses in the different spatial directions are summed algebraically by

q_{α}^{m a x} = \sum_{k} {(q_{α}^{m a x})}_{k} .

After this summation is performed, the modal responses are summed. (Choosing the method used for modal summation is described below in Modal Summation Methods.) Since the directional components are summed first, the subscript k is not relevant and can be ignored in the modal summation equations that follow.

Input File Usage

RESPONSE SPECTRUM, COMP=ALGEBRAIC, SUM=sum

Abaqus/CAE Usage

Step module: Create Step: Linear perturbation: Response spectrum: Excitations: Single direction or Multiple direction absolute sum

The Square Root of the Sum of the Squares Directional Summation Method

If the spectra in different directions represent independent excitations, the modal summation is performed first, as explained below in Modal Summation Methods. Then, the estimates of the total peak response in each excitation direction ${(R^{i})}_{k}^{m a x}$ are combined by

{(R^{i})}^{m a x} = \sqrt{\sum_{k} {({(R^{i})}_{k}^{m a x})}^{2}} .

Input File Usage

RESPONSE SPECTRUM, COMP=SRSS, SUM=sum

Abaqus/CAE Usage

Step module: Create Step: Linear perturbation: Response spectrum: Excitations: Multiple direction square root of the sum of squares

The Forty-Percent Method

If the spectra in different directions represent independent excitations, the modal summation is performed first, as explained below in Modal Summation Methods. Then, the responses in different excitation directions are combined by the 40% rule recommended by the ASCE 4–98 standard for Seismic Analysis of Safety-Related Nuclear Structures and Commentary, Section 3.2.7.1.2. This method combines the response for all possible combinations of the three components, including variations in sign (plus/minus), assuming that when the maximum response from one component occurs, the response from the other two components is 40% of their maximum value, using one of the following:

{(R^{i})}^{m a x} = \pm [{(R^{i})}_{1}^{m a x} \pm 0.4 {(R^{i})}_{2}^{m a x} \pm 0.4 {(R^{i})}_{3}^{m a x}],

{(R^{i})}^{m a x} = \pm [{(R^{i})}_{2}^{m a x} \pm 0.4 {(R^{i})}_{1}^{m a x} \pm 0.4 {(R^{i})}_{3}^{m a x}],

{(R^{i})}^{m a x} = \pm [{(R^{i})}_{3}^{m a x} \pm 0.4 {(R^{i})}_{2}^{m a x} \pm 0.4 {(R^{i})}_{1}^{m a x}] .

Input File Usage

RESPONSE SPECTRUM, COMP=R40, SUM=sum

Abaqus/CAE Usage

Step module: Create Step: Linear perturbation: Response spectrum: Excitations: Multiple direction forty percent rule

The Thirty-Percent Method

If the spectra in different directions represent independent excitations, the modal summation is performed first, as explained below in Modal Summation Methods. Then, the responses in different excitation directions are combined by the 30% rule recommended by the ASCE 4–98 standard for Seismic Analysis of Safety-Related Nuclear Structures and Commentary, Section 3.2.7.1.2. This method combines the response for all possible combinations of the three components, including variations in sign (plus/minus), assuming that when the maximum response from one component occurs, the response from the other two components is 30% of their maximum value, using one of the following:

{(R^{i})}^{m a x} = \pm [{(R^{i})}_{1}^{m a x} \pm 0.3 {(R^{i})}_{2}^{m a x} \pm 0.3 {(R^{i})}_{3}^{m a x}],

{(R^{i})}^{m a x} = \pm [{(R^{i})}_{2}^{m a x} \pm 0.3 {(R^{i})}_{1}^{m a x} \pm 0.3 {(R^{i})}_{3}^{m a x}],

{(R^{i})}^{m a x} = \pm [{(R^{i})}_{3}^{m a x} \pm 0.3 {(R^{i})}_{2}^{m a x} \pm 0.3 {(R^{i})}_{1}^{m a x}] .

Input File Usage

RESPONSE SPECTRUM, COMP=R30, SUM=sum

Abaqus/CAE Usage

Step module: Create Step: Linear perturbation: Response spectrum: Excitations: Multiple direction thirty percent rule

Modal Summation Methods

The peak response of some physical variable $R^{i}$ (a component i of displacement, stress, section force, reaction force, etc.) caused by the motion in the $α$ th natural mode excited by the given response spectra in direction k at frequency $ω_{α}$ with damping $ξ_{α}$ is given by

{(R_{α}^{i})}_{k}^{m a x} = Φ_{α}^{i} {(q_{α}^{m a x})}_{k},

where $Φ_{α}^{i}$ is the ith component of mode $α$ , and there is no sum on $α$ . (In the case of algebraic summation the subscript k is not relevant and can be ignored in this equation and in those that follow.)

There are several methods for combining these peak physical responses in the individual modes, ${(R_{α}^{i})}_{k}^{m a x}$ , into estimates of the total peak response, ${(R^{i})}_{k}^{m a x}$ . Most of the methods implemented in Abaqus/Standard follow the ASCE 4–98 standard for Seismic Analysis of Safety Related Nuclear Structures and Commentary. The updated documents, “Reevaluation of Regulatory Guidance on Modal Response Combination Methods for Seismic Response Spectrum Analysis” issued in 1999 (NUREG/CR-6645, BNL-NUREG-52276) and “Draft Regulatory Guide” (DG-1127) issued in 2005 contain new recommendations. You are advised to read the new recommendations before choosing a modal summation method from among those described below.

The Absolute Value Method

The absolute value method is the most conservative method for combining the modal responses. It is obtained by summing the absolute values resulting from each mode:

{(R^{i})}_{k}^{m a x} = \sum_{α} | {(R_{α}^{i})}_{k}^{m a x} | .

This method implies that all of the responses peak simultaneously. It will overpredict the peak response of most systems; therefore, it may be too conservative to help in design.

Input File Usage

RESPONSE SPECTRUM, COMP=comp, SUM=ABS

Abaqus/CAE Usage

Step module: Create Step: Linear perturbation: Response spectrum: Summations: Absolute values

The Square Root of the Sum of the Squares Modal Summation Method

The square root of the sum of the squares method is less conservative than the absolute value method. It is also usually more accurate if the natural frequencies of the system are well separated. It uses the square root of the sum of the squares to combine the modal responses:

{(R^{i})}_{k}^{m a x} = \sqrt{\sum_{α} {({(R_{α}^{i})}_{k}^{m a x})}^{2}} .

Input File Usage

RESPONSE SPECTRUM, COMP=comp, SUM=SRSS

Abaqus/CAE Usage

Step module: Create Step: Linear perturbation: Response spectrum: Summations: Square root of the sum of squares

The Naval Research Laboratory Method

The absolute value and square root of the sum of the squares methods can be combined to yield the Naval Research Laboratory method. It distinguishes the mode, $β$ , in which the physical variable has its maximum response and adds the square root of the sum of squares of the peak responses in all other modes to the absolute value of the peak response of that mode. This method gives the estimate:

{(R^{i})}_{k}^{m a x} = | {(R_{β}^{i})}_{k}^{m a x} | + \sqrt{\sum_{α \neq β} {({(R_{α}^{i})}_{k}^{m a x})}^{2}} .

Input File Usage

RESPONSE SPECTRUM, COMP=comp, SUM=NRL

Abaqus/CAE Usage

Step module: Create Step: Linear perturbation: Response spectrum: Summations: Naval Research Laboratory

The Ten-Percent Method

The ten-percent method recommended by Regulatory Guide 1.92 (1976) is no longer recommended according to the “Reevaluation of Regulatory Guidance on Modal Response Combination Methods for Seismic Response Spectrum Analysis” document issued in 1999. It is retained here because of its extensive prior use. The ten-percent method modifies the square root of the sum of the squares method by adding a contribution from all pairs of modes $α$ and $β$ whose frequencies are within 10% of each other, giving the estimate:

{(R^{i})}_{k}^{m a x} = \sqrt{\sum_{α} {({(R_{α}^{i})}_{k}^{m a x})}^{2} + 2 \sum_{α < β} | {(R_{α}^{i})}_{k}^{m a x} {(R_{β}^{i})}_{k}^{m a x} |} .

The frequencies of modes $α$ and $β$ are considered to be within 10% of each other whenever

\frac{ω_{β} - ω_{α}}{ω_{β}} \leq 0.1, α < β .

The ten-percent method reduces to the square root of the sum of the squares method if the modes are well separated with no coupling between them.

Input File Usage

RESPONSE SPECTRUM, COMP=comp, SUM=TENP

Abaqus/CAE Usage

Step module: Create Step: Linear perturbation: Response spectrum: Summations: Ten percent

The Complete Quadratic Combination Method

Like the ten-percent method, the complete quadratic combination method improves the estimation for structures with closely spaced eigenvalues. The complete quadratic combination method combines the modal response with the formula

{(R^{i})}_{k}^{m a x} = \sqrt{\sum_{α} \sum_{β} {(R_{α}^{i})}_{k}^{m a x} ρ_{α β} {(R_{β}^{i})}_{k}^{m a x}},

where $ρ_{α β}$ are cross-correlation coefficients between modes $α$ and $β$ , which depend on the ratio of frequencies and modal damping between the two modes:

ρ_{α β} = \frac{8 \sqrt{ξ_{α} ξ_{β}} (ξ_{α} + r_{β α} ξ_{β}) r_{β α}^{3 / 2}}{{(1 - r_{β α}^{2})}^{2} + 4 ξ_{α} ξ_{β} r_{β α} (1 + r_{β α}^{2}) + 4 (ξ_{α}^{2} + ξ_{β}^{2}) r_{β α}^{2}},

where $r_{β α} = ω_{β} / ω_{α}$ .

If the modes are well spaced, their cross-correlation coefficient will be small ( $ρ_{α β} ≪ 1$ ) and the method will give the same results as the square root of the sum of the squares method.

This method is usually recommended for asymmetrical building systems since, in such cases, other methods can underestimate the response in the direction of motion and overestimate the response in the transverse direction.

Input File Usage

RESPONSE SPECTRUM, COMP=comp, SUM=CQC

Abaqus/CAE Usage

Step module: Create Step: Linear perturbation: Response spectrum: Summations: Complete quadratic combination

The Grouping Method

This method, also known as the NRC grouping method, improves the response estimation for structures with closely spaced eigenvalues. The modal responses are grouped such that the lowest and highest frequency modes in a group are within 10% and no mode is in more than one group. The modal responses are summed absolutely within groups before performing a SRSS combination of the groups. Within the group responses are summed as

{(R_{g r}^{i})}_{k}^{m a x} = | {(R_{1}^{i})}_{k}^{m a x} | + | {(R_{2}^{i})}_{k}^{m a x} | \dots + | {(R_{n}^{i})}_{k}^{m a x} |,

for “n” frequencies within any “gr” group and then performing

{(R^{i})}_{k}^{m a x} = \sqrt{\overset{g r = n}{\sum_{g r = 1}} {({(R_{g r}^{i})}_{k}^{max})}^{2}} .

The above expression includes all the groups; in addition, the group can consist of just one frequency response if this frequency does not have another member that is within the 10% limit.

The ten-percent method will always produce results higher in value than the grouping method.

Input File Usage

RESPONSE SPECTRUM, COMP=comp, SUM=GRP

Abaqus/CAE Usage

Step module: Create Step: Linear perturbation: Response spectrum: Summations: Grouping method

Double Sum Combination

This method, also known as Rosenblueth's double sum combination (Rosenblueth and Elorduy, 1969), is the first attempt to evaluate modal correlation based on random vibration theory. It utilizes the time duration $t_{D}$ of strong earthquake motion. The mode correlation coefficients $c_{α β}$ , which depend also on the frequencies and damping coefficient $ξ$ , lead to the following mode combination:

{(R^{i})}_{k}^{m a x} = \sqrt{\sum_{α} \sum_{β} {(R_{α}^{i})}_{k}^{m a x} c_{α β} {(R_{β}^{i})}_{k}^{m a x}},

where

c_{α β} = \frac{1}{1 + {(\frac{ω_{α^{'}} - ω_{β^{'}}}{ξ_{α^{'}} ω_{α} + ξ_{β^{'}} ω_{β}})}^{2}},

where

ω_{α^{'}} = ω_{α} \sqrt{1 - ξ_{α}^{2}},

ξ_{α^{'}} = ξ_{α} + \frac{2}{t_{D} ω_{α}} .

Input File Usage

RESPONSE SPECTRUM, COMP=comp, SUM=DSC

Abaqus/CAE Usage

Step module: Create Step: Linear perturbation: Response spectrum: Summations: Double sum combination

Separation of Modal Responses into Periodic and Rigid Responses

Each spectrum can be divided into low-, medium-, and high-frequency regions. In the low-frequency region the modes are usually uncorrelated. In the mid-range they are partially correlated, and in the high-frequency region they tend to be correlated with the input data and, hence, between themselves. Each of these regions may, therefore, require special consideration. These considerations are addressed by the Gupta and Lindley-Yow methods.

We divide the modal response into two parts: the rigid part and the damped periodic part. It is assumed that the rigid part and the damped periodic part are statistically independent. Let us define the rigid response coefficient for each mode by $0 \leq α_{β} \leq 1$ , the rigid response by ${(R_{β}^{i})}^{r} {= α}_{β} (R_{β}^{i})$ , and the periodic response by ${(R_{β}^{i})}^{p} = R_{β}^{i} \sqrt{1 - α_{β}^{2}}^{}$ . Therefore, the relationship between both regions becomes:

{({(R_{β}^{i})}^{p})}^{2} + {({(R_{β}^{i})}^{r})}^{2} = 1,

which is equivalent to summing both regions according to the SRSS rule. The value of the rigid response coefficient is chosen depending on the method used. For the Gupta method this coefficient is estimated based on two user-specified frequencies: $F_{1}$ defining the beginning of the periodic region, and $F_{2}$ defining the beginning of the rigid region (which is often equivalent to the frequency associated with zero period acceleration). For the current frequency $F_{β}$ , this expression for the Gupta method is

α_{β} = \frac{\log (\frac{F_{β}}{F_{1}})}{\log (\frac{F_{2}}{F_{1}})} .

For the Lindley-Yow method this coefficient is equal to

α_{β} = \frac{Z P A}{S_{a β}},

where ZPA denotes the zero period acceleration for the user-specified frequency value and $S_{a β}$ denotes the acceleration spectrum for mode $β$ . The Lindley-Yow method does not work well for the region of frequencies on the low-frequency end of the spectrum. Therefore, we set the coefficient $α_{β} = 0$ for all modes below the frequency of the peak acceleration value for this spectrum.

The rigid response method works with only two modal summation rules: CQC and DSC with their respective definitions for the mode correlation coefficients $c_{α β}$ . By default, no rigid response is included. To include a rigid response, you must specify the rigid response method.

Input File Usage

Use the following option to specify the Gupta method for the rigid response:

RESPONSE SPECTRUM, COMP=comp, RIGID RESPONSE=GUPTA

Use the following option to specify the Lindley-Yow method for the rigid response:

RESPONSE SPECTRUM, COMP=comp, RIGID RESPONSE=LINDLEY-YOW

Abaqus/CAE Usage

You cannot specify a rigid response in Abaqus/CAE.

Selecting the Modes and Specifying Damping

You can select the modes to be used in modal superposition and specify damping values for all selected modes.

Selecting the Modes

You can select modes by specifying the mode numbers individually, by requesting that Abaqus/Standard generate the mode numbers automatically, or by requesting the modes that belong to specified frequency ranges. If you do not select the modes, all modes extracted in the prior eigenfrequency extraction step, including residual modes if they were activated, are used in the modal superposition.

Input File Usage

Use one of the following options to select the modes by specifying mode numbers:

SELECT EIGENMODES, DEFINITION=MODE NUMBERS
SELECT EIGENMODES, GENERATE, DEFINITION=MODE NUMBERS

Use the following option to select the modes by specifying a frequency range:

SELECT EIGENMODES, DEFINITION=FREQUENCY RANGE

Abaqus/CAE Usage

You cannot select the modes in Abaqus/CAE; all modes extracted are used in the modal superposition.

Specifying Damping

Damping is almost always specified for a mode-based procedure; see Material Damping. You can define a damping coefficient for all or some of the modes used in the response calculation. The damping coefficient can be given for a specified mode number or for a specified frequency range. When damping is defined by specifying a frequency range, the damping coefficient for an mode is interpolated linearly between the specified frequencies. The frequency range can be discontinuous; the average damping value will be applied for an eigenfrequency at a discontinuity. The damping coefficients are assumed to be constant outside the range of specified frequencies.

Input File Usage

Use the following option to define damping by specifying mode numbers:

MODAL DAMPING, DEFINITION=MODE NUMBERS

Use the following option to define damping by specifying a frequency range:

MODAL DAMPING, DEFINITION=FREQUENCY RANGE

Abaqus/CAE Usage

Use the following input to define damping by specifying mode numbers:

Step module: Create Step: Linear perturbation: Response spectrum: 
Damping: Specify damping over ranges of: Modes

Use the following input to define damping by specifying a frequency range:

Step module: Create Step: Linear perturbation: Response spectrum: 
Damping: Specify damping over ranges of: Frequencies

Example of Specifying Damping

Figure 1 illustrates how the damping coefficients at different eigenfrequencies are determined for the following input:

MODAL DAMPING, DEFINITION=FREQUENCY RANGE
 $f_{1}, d_{1}$ 
 $f_{2}, d_{2}$ 
 $f_{2}, d_{3}$ 
 $f_{3}, d_{3}$ 
 $f_{4}, d_{4}$

Rules for Selecting Modes and Specifying Damping Coefficients

The following rules apply for selecting modes and specifying modal damping coefficients:

No modal damping is included by default.
Mode selection and modal damping must be specified in the same way, using either mode numbers or a frequency range.
If you do not select any modes, all modes extracted in the prior frequency analysis, including residual modes if they were activated, will be used in the superposition.
If you do not specify damping coefficients for modes that you have selected, zero damping values will be used for these modes.
Damping is applied only to the modes that are selected.
Damping coefficients for selected modes that are beyond the specified frequency range are constant and equal to the damping coefficient specified for the first or the last frequency (depending which one is closer). This is consistent with the way Abaqus interprets amplitude definitions.

Using the Missing Mass Method

You can use the missing mass method to augment the missing inertia in the dynamic response when the number of eigenmodes included in the superposition does not sufficiently represent the total mass of the structure. The missing inertia is due primarily to the truncated higher frequency modes that belong to the in-phase (rigid) response. Quasistatic residual modes in the direction of applied spectra can be substituted for these truncated modes. This technique is most suitable for use with the response spectra summations that split the frequency range into periodic and rigid phases, such as the Gupta or Lindley-Yow methods. However, any modal summation method can be used with the missing mass method. The missing mass method results in better response accuracy and is explained in three small spring-mass examples introduced first in the paper by Dhileep (Dhileep and Bose, 2008). The residual vector representing missing energy is found by a simple inertia balance in eigenspace:

K x_{0} = - M (u_{B} - \sum_{i = 1}^{m} Φ_{i} Γ_{i}) a_{g},

where

$K$: represents the stiffness of the model,
$M$: represents the mass of the model,
$Φ_{i}$ and $Γ_{i}$: represent the $i^{t h}$ out of $m$ eigenvectors and participation factors,
$u_{B}$: is the unit base displacement in the direction of the earthquake spectrum,
$a_{g}$: is a ground acceleration, and
$x_{0}$: is a pseudostatic residual mode representing the response of the remaining modes beyond the mode $m$ .

Input File Usage

RESPONSE SPECTRUM, MISSING MASS METHOD

Abaqus/CAE Usage

The missing mass method is not supported in Abaqus/CAE.

Initial Conditions

It is not appropriate to specify initial conditions in a response spectrum analysis.

Boundary Conditions

All points constrained by boundary conditions and the ground nodes of connector elements are assumed to move in phase in any one direction. This base motion can use a different input spectrum in each of three orthogonal directions (two directions in a two-dimensional model). You define the input spectra, $S (ω, ξ)$ , as functions of frequency, $ω$ , for different values of critical damping, $ξ$ , as described earlier in Specifying a Spectrum. Secondary bases cannot be used in a response spectrum analysis.

Loads

The only “loading” that can be defined in a response spectrum analysis is that defined by the input spectra, as described earlier. No other loads can be prescribed in a response spectrum analysis.

Predefined Fields

Predefined fields, including temperature, cannot be used in response spectrum analysis.

Material Options

The density of the material must be defined (Density). The following material properties are not active during a response spectrum analysis: plasticity and other inelastic effects, rate-dependent material properties, thermal properties, mass diffusion properties, electrical properties, and pore fluid flow properties—see General and Perturbation Procedures.

Elements

Other than generalized axisymmetric elements with twist, any of the stress/displacement elements in Abaqus/Standard can be used in a response spectrum analysis—see Choosing the Appropriate Element for an Analysis Type.

Output

All the output variables in Abaqus/Standard are listed in Abaqus/Standard Output Variable Identifiers. The value of an output variable such as strain, E; stress, S; or displacement, U, is its peak magnitude.

In addition to the usual output variables available, the following modal variables are available for response spectrum analysis and can be output to the data and/or results files (see Output to the Data and Results Files):

GU: Generalized displacements for all modes.
GV: Generalized velocities for all modes.
GA: Generalized accelerations for all modes.
SNE: Elastic strain energy for the entire model per each mode.
KE: Kinetic energy for the entire model per each mode.
T: External work for the entire model per each mode.

Neither element energy densities (such as the elastic strain energy density, SENER) nor whole element energies (such as the total kinetic energy of an element, ELKE) are available for output in response spectrum analysis. However, whole model variables such as ALLIE (total strain energy) are available for modal-based procedures as output to the data and/or results files (see Output to the Data and Results Files).

Reaction force output is not supported for response spectrum analysis using eigenmodes extracted using the default SIM-based frequency extraction procedure with either the AMS or Lanczos eigensolver. Reaction force output in response spectrum analysis using eigenmodes extracted with the non-SIM-based Lanczos eigensolver provides directional combinations of so-called, modal reaction forces weighted with maximal absolute values of corresponding generalized displacements. Directional and modal combination rules used for the reaction force calculation are the same as for other nodal output variables. Modal reaction forces are calculated in the frequency extraction procedure. They represent static reaction forces calculated for the normal mode shapes. Generally, they cannot adequately represent reaction force in dynamic analysis. For models with diagonal mass and diagonal damping matrices the superposition of the modal reaction forces can provide a reasonable approximation of a nodal reaction force in mode-based analyses other than response spectrum analysis. In response spectrum analysis the model response can be better represented by requesting section stresses and section forces in structural elements containing supported nodes.

Input File Template

HEADING
…
BOUNDARY
Data lines to define points to be excited by the base motion controlled by the input spectra
SPECTRUM, NAME=name1, TYPE=type
Data lines to define spectrum “name1” as a function of frequency,  $ω$ , and
fraction of critical damping,  $ξ$ 
SPECTRUM, NAME=name2, TYPE=type
Data lines to define spectrum “name2” as a function of frequency,  $ω$ , and
fraction of critical damping,  $ξ$ 
**
STEP
FREQUENCY
Data line to specify number of modes to be extracted
END STEP
**
STEP
RESPONSE SPECTRUM, COMP=comp, SUM=sum
Data lines referring to response spectra and defining direction cosines
SELECT EIGENMODES
Data lines to define the applicable mode ranges
MODAL DAMPING
Data lines to define modal damping
END STEP

References

Dhileep, M., and P. R. Bose, “A Comparative Study of "Missing Mass" Correction Methods for Response Spectrum Method of Seismic Analysis,” Computers and Structures, vol. 86, pp. 2087–2094, 2008.
Rosenblueth, E., and J. Elorduy, “Response of Linear Systems to Certain Transient Disturbances,” Proceedings of the Fourth World Conference on Earthquake Engineering, Santiago, Chile, 1969.